Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 2

For a regular scalar statistical model, the one-observation Fisher information isUnder the usual differentiation and interchange-of-integral conditions it also equals $-{\mathbb{E}}_{\theta }[{\partial }_{\theta }^2\log p(Y\mid \theta )]$. The Jeffreys prior isThis prior measure is invariant under smooth one-to-one reparameterization, but may be improper. For independent identically distributed observations the information multiplier $n$ changes only the prior's proportionality constant.

For the scale family, substitute $u=y/\sigma$ in the expectation:This depends on $f,g$ but not on $\sigma$. It holds whenever the expectation exists, including an extended expectation for nonnegative $g$; an undefined difference of two infinite integrals is not an expectation.

Assume the scale family has the differentiability needed for Fisher information, and the following constant is finite and positive. With ${\ell }_f(u)=\log f(u)$, the score function isPart (b) therefore giveswhere $C$ is independent of $\sigma$. The Jeffreys prior for a scale parameter is consequentlyThe reciprocal-of-a-reciprocal in the TeX is a transcription error; the PDF has ${\sigma }^{-1}$. The expected-Hessian information identity should not be applied indiscriminately to nonregular parameter-dependent supports.

For $\tau =c\sigma$ with $c>0$, the change of variables givesEquivalently,This is the scale-invariant prior as a measure. Because its integral over $(0,\infty )$ diverges, it is an improper prior, not a normalized probability distribution on that range.

The normalized log-uniform distribution here has density $1/(x\log 10)$. A leading digit $i$ corresponds to $i\le X<i+1$, soThese are exactly the Benford law probabilities. Continuous densities make endpoint conventions immaterial.

The claim needs a complete number of logarithmic decades. For the normalized log-uniform distribution on the specified range, $U={\log }_{10}X$ is uniform on $(a,b)$. Digit $i$ corresponds to $\{U\}\in [{\log }_{10}i,{\log }_{10}(i+1))$. If $b-a=m$ is a positive integer, the integral of a period-one indicator over $(a,a+m)$ is $m$ times its integral over one period. Thus Benford law from logarithmic uniformity givesThis proves the intended case of integers $a<b$, and even permits noninteger $a$ when the span is an integer.

For arbitrary real $a<b$, the exact formula instead isOnly finitely many terms are nonzero. For a counterexample take $a=0$, $b={\log }_{10}2$: then $1<X<2$, so the leading digit is always one. That is not Benford law. The unrestricted range in the PDF needs this qualification.

A predictive discrepancy statistic is a specified measurable function $T$ of a possible dataset, chosen to detect a feature relevant to the null hypothesis. Under the fully specified null density $p_0$, compare the observed $T(\mathbf{x})$ with its reference distribution, derived analytically or from replicated datasets ${\mathbf{X}}^{\mathrm{rep}}\sim p_0$.

For discrepancies whose large values indicate disagreement, useA lower-tail or two-sided discrepancy requires the corresponding comparison. The checking function is a statistic, not a density; no unknown parameter is fitted when the null is fully specified.

Assuming independent digits under Benford law, put $N={\sum }_i{y}_i$ and $p_i={\log }_{10}(1+1/i)$. The count vector has a multinomial distribution, with expected counts $E_i=N{p}_i$. Suitable predictive discrepancy statistics include the Pearson chi-squared statistic and multinomial deviance:The zero-count terms of $T_G$ have limiting value zero. A Monte Carlo method gives a direct null comparison even when expected counts are small.

For example, an original R implementation is:Each column returned by rmultinom is a replicated count vector. R recycles the nine expected counts down each column. The add-one ratio is a Monte Carlo test estimate. In WinBUGS, alternatively generate a replicated dmulti vector using fixed $p_i,N$, compute its discrepancy and monitor exceedance of the observed value. The null does not estimate unknown digit probabilities. Dependence or selection in the accounts would require an appropriate simulation model.

Changing currency or monetary units multiplies amounts by a common constant without changing their generating mechanism. This motivates scale invariance of decimal significands of a generic significand distribution.

Multiplication by $c$ adds ${\log }_{10}c$ to the logarithm, rotating its fractional part. A probability distribution on the unit circle invariant under every rotation must be uniform: equal-length arcs have equal mass, and subdivision fixes that mass to arc length. Uniform fractional logarithms then give Benford law. Thus unit or currency invariance motivates uniform logarithmic mantissas. Restricted ranges, prescribed thresholds and rounding can prevent a particular collection from having this invariance.

Yes, the pattern merits checking. Digits two and three are visibly more frequent than the Benford law predictions, whereas several large digits are scarce and nine is absent. Rough agreement for one and four does not remove this pattern.

Use the predictive discrepancy statistic comparison in part (h), together with examination of the accounts' selection and constraints. A discrepancy is evidence against this digit model, not by itself evidence sufficient to establish fabrication. No numerical test calculation is needed for this qualitative assessment.